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G = C7⋊C32order 224 = 25·7

The semidirect product of C7 and C32 acting via C32/C16=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C7⋊C32, C14.C16, C56.2C4, C28.2C8, C16.2D7, C112.3C2, C8.3Dic7, C2.(C7⋊C16), C4.2(C7⋊C8), SmallGroup(224,1)

Series: Derived Chief Lower central Upper central

C1C7 — C7⋊C32
C1C7C14C28C56C112 — C7⋊C32
C7 — C7⋊C32
C1C16

Generators and relations for C7⋊C32
 G = < a,b | a7=b32=1, bab-1=a-1 >

7C32

Smallest permutation representation of C7⋊C32
Regular action on 224 points
Generators in S224
(1 170 142 193 78 110 33)(2 34 111 79 194 143 171)(3 172 144 195 80 112 35)(4 36 113 81 196 145 173)(5 174 146 197 82 114 37)(6 38 115 83 198 147 175)(7 176 148 199 84 116 39)(8 40 117 85 200 149 177)(9 178 150 201 86 118 41)(10 42 119 87 202 151 179)(11 180 152 203 88 120 43)(12 44 121 89 204 153 181)(13 182 154 205 90 122 45)(14 46 123 91 206 155 183)(15 184 156 207 92 124 47)(16 48 125 93 208 157 185)(17 186 158 209 94 126 49)(18 50 127 95 210 159 187)(19 188 160 211 96 128 51)(20 52 97 65 212 129 189)(21 190 130 213 66 98 53)(22 54 99 67 214 131 191)(23 192 132 215 68 100 55)(24 56 101 69 216 133 161)(25 162 134 217 70 102 57)(26 58 103 71 218 135 163)(27 164 136 219 72 104 59)(28 60 105 73 220 137 165)(29 166 138 221 74 106 61)(30 62 107 75 222 139 167)(31 168 140 223 76 108 63)(32 64 109 77 224 141 169)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)

G:=sub<Sym(224)| (1,170,142,193,78,110,33)(2,34,111,79,194,143,171)(3,172,144,195,80,112,35)(4,36,113,81,196,145,173)(5,174,146,197,82,114,37)(6,38,115,83,198,147,175)(7,176,148,199,84,116,39)(8,40,117,85,200,149,177)(9,178,150,201,86,118,41)(10,42,119,87,202,151,179)(11,180,152,203,88,120,43)(12,44,121,89,204,153,181)(13,182,154,205,90,122,45)(14,46,123,91,206,155,183)(15,184,156,207,92,124,47)(16,48,125,93,208,157,185)(17,186,158,209,94,126,49)(18,50,127,95,210,159,187)(19,188,160,211,96,128,51)(20,52,97,65,212,129,189)(21,190,130,213,66,98,53)(22,54,99,67,214,131,191)(23,192,132,215,68,100,55)(24,56,101,69,216,133,161)(25,162,134,217,70,102,57)(26,58,103,71,218,135,163)(27,164,136,219,72,104,59)(28,60,105,73,220,137,165)(29,166,138,221,74,106,61)(30,62,107,75,222,139,167)(31,168,140,223,76,108,63)(32,64,109,77,224,141,169), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)>;

G:=Group( (1,170,142,193,78,110,33)(2,34,111,79,194,143,171)(3,172,144,195,80,112,35)(4,36,113,81,196,145,173)(5,174,146,197,82,114,37)(6,38,115,83,198,147,175)(7,176,148,199,84,116,39)(8,40,117,85,200,149,177)(9,178,150,201,86,118,41)(10,42,119,87,202,151,179)(11,180,152,203,88,120,43)(12,44,121,89,204,153,181)(13,182,154,205,90,122,45)(14,46,123,91,206,155,183)(15,184,156,207,92,124,47)(16,48,125,93,208,157,185)(17,186,158,209,94,126,49)(18,50,127,95,210,159,187)(19,188,160,211,96,128,51)(20,52,97,65,212,129,189)(21,190,130,213,66,98,53)(22,54,99,67,214,131,191)(23,192,132,215,68,100,55)(24,56,101,69,216,133,161)(25,162,134,217,70,102,57)(26,58,103,71,218,135,163)(27,164,136,219,72,104,59)(28,60,105,73,220,137,165)(29,166,138,221,74,106,61)(30,62,107,75,222,139,167)(31,168,140,223,76,108,63)(32,64,109,77,224,141,169), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224) );

G=PermutationGroup([(1,170,142,193,78,110,33),(2,34,111,79,194,143,171),(3,172,144,195,80,112,35),(4,36,113,81,196,145,173),(5,174,146,197,82,114,37),(6,38,115,83,198,147,175),(7,176,148,199,84,116,39),(8,40,117,85,200,149,177),(9,178,150,201,86,118,41),(10,42,119,87,202,151,179),(11,180,152,203,88,120,43),(12,44,121,89,204,153,181),(13,182,154,205,90,122,45),(14,46,123,91,206,155,183),(15,184,156,207,92,124,47),(16,48,125,93,208,157,185),(17,186,158,209,94,126,49),(18,50,127,95,210,159,187),(19,188,160,211,96,128,51),(20,52,97,65,212,129,189),(21,190,130,213,66,98,53),(22,54,99,67,214,131,191),(23,192,132,215,68,100,55),(24,56,101,69,216,133,161),(25,162,134,217,70,102,57),(26,58,103,71,218,135,163),(27,164,136,219,72,104,59),(28,60,105,73,220,137,165),(29,166,138,221,74,106,61),(30,62,107,75,222,139,167),(31,168,140,223,76,108,63),(32,64,109,77,224,141,169)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)])

C7⋊C32 is a maximal subgroup of   D7×C32  C32⋊D7  C7⋊M6(2)  C7⋊D32  D16.D7  C7⋊SD64  C7⋊Q64
C7⋊C32 is a maximal quotient of   C7⋊C64

80 conjugacy classes

class 1  2 4A4B7A7B7C8A8B8C8D14A14B14C16A···16H28A···28F32A···32P56A···56L112A···112X
order1244777888814141416···1628···2832···3256···56112···112
size111122211112221···12···27···72···22···2

80 irreducible representations

dim11111122222
type+++-
imageC1C2C4C8C16C32D7Dic7C7⋊C8C7⋊C16C7⋊C32
kernelC7⋊C32C112C56C28C14C7C16C8C4C2C1
# reps11248163361224

Matrix representation of C7⋊C32 in GL3(𝔽449) generated by

100
04481
039652
,
40400
05211
0223444
G:=sub<GL(3,GF(449))| [1,0,0,0,448,396,0,1,52],[404,0,0,0,5,223,0,211,444] >;

C7⋊C32 in GAP, Magma, Sage, TeX

C_7\rtimes C_{32}
% in TeX

G:=Group("C7:C32");
// GroupNames label

G:=SmallGroup(224,1);
// by ID

G=gap.SmallGroup(224,1);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,12,31,50,69,6917]);
// Polycyclic

G:=Group<a,b|a^7=b^32=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C7⋊C32 in TeX

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